Free piston Stirling engine (FPSE) is a closed cycle engine that converts thermal energy into mechanical energy. The focus of this thesis is on understanding limit-cycle motions in FPSEs. First, making use of reduced-order models, parametric studies are carried out to understand what FPSE parameters affect the creation of oscillatory motions. It is shown that quasi-static variations of the stiffness and the damping terms on the power piston can lead to conditions for Hopf instabilities in the system. The effect of the inclusion of a nonlinear spring term to the system is also investigated through numerical studies. The nonlinear springs include hardening springs and magnetic springs. The results, which include first results for FPSEs with hardening springs, show that nonlinear springs can help these systems realize limit-cycle motions. Finally, preliminary experimental studies conducted to realize oscillatory motions in a FPSE with a magnetic spring are also reported.